dgges(3S)
DGGES - compute for a pair of N-by-N real nonsymmetric matrices (A,B),
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NAME DGGES - compute for a pair of N-by-N real nonsymmetric matrices (A,B), SYNOPSIS SUBROUTINE DGGES( JOBVSL, JOBVSR, SORT, DELCTG, N, A, LDA, B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, BWORK, INFO ) CHARACTER JOBVSL, JOBVSR, SORT INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM LOGICAL BWORK( * ) DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ), BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ), WORK( * ) LOGICAL DELCTG EXTERNAL DELCTG IMPLEMENTATION These routines are part of the SCSL Scientific Library and can be loaded using either the -lscs or the -lscs_mp option. The -lscs_mp option directs the linker to use the multi-processor version of the library. When linking to SCSL with -lscs or -lscs_mp, the default integer size is 4 bytes (32 bits). Another version of SCSL is available in which integers are 8 bytes (64 bits). This version allows the user access to larger memory sizes and helps when porting legacy Cray codes. It can be loaded by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use only one of the two versions; 4-byte integer and 8-byte integer library calls cannot be mixed. PURPOSE DGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B), the generalized eigenvalues, the generalized real Schur form (S,T), optionally, the left and/or right matrices of Schur vectors (VSL and VSR). This gives the generalized Schur factorization (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T ) Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper quasi- triangular matrix S and the upper triangular matrix T.The leading columns of VSL and VSR then form an orthonormal basis for the corresponding left and right eigenspaces (deflating subspaces). (If only the generalized eigenvalues are needed, use the driver DGGEV instead, which is faster.) A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha/beta = w, such that A - w*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0 or both being zero. A pair of matrices (S,T) is in generalized real Schur form if T is upper triangular with non-negative diagonal and S is block upper triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond to real generalized eigenvalues, while 2-by-2 blocks of S will be "standardized" by making the corresponding elements of T have the form: [ a 0 ] [ 0 b ] and the pair of corresponding 2-by-2 blocks in S and T will have a complex conjugate pair of generalized eigenvalues. ARGUMENTS JOBVSL (input) CHARACTER*1 = 'N': do not compute the left Schur vectors; = 'V': compute the left Schur vectors. JOBVSR (input) CHARACTER*1 = 'N': do not compute the right Schur vectors; = 'V': compute the right Schur vectors. SORT (input) CHARACTER*1 Specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form. = 'N': Eigenvalues are not ordered; = 'S': Eigenvalues are ordered (see DELCTG); DELCTG (input) LOGICAL FUNCTION of three DOUBLE PRECISION arguments DELCTG must be declared EXTERNAL in the calling subroutine. If SORT = 'N', DELCTG is not referenced. If SORT = 'S', DELCTG is used to select eigenvalues to sort to the top left of the Schur form. An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if DELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either one of a complex conjugate pair of eigenvalues is selected, then both complex eigenvalues are selected. Note that in the ill-conditioned case, a selected complex eigenvalue may no longer satisfy DELCTG(ALPHAR(j),ALPHAI(j), BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2 in this case. N (input) INTEGER The order of the matrices A, B, VSL, and VSR. N >= 0. A (input/output) DOUBLE PRECISION array, dimension (LDA, N) On entry, the first of the pair of matrices. On exit, A has been overwritten by its generalized Schur form S. LDA (input) INTEGER The leading dimension of A. LDA >= max(1,N). B (input/output) DOUBLE PRECISION array, dimension (LDB, N) On entry, the second of the pair of matrices. On exit, B has been overwritten by its generalized Schur form T. LDB (input) INTEGER The leading dimension of B. LDB >= max(1,N). SDIM (output) INTEGER If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM = number of eigenvalues (after sorting) for which DELCTG is true. (Complex conjugate pairs for which DELCTG is true for either eigenvalue count as 2.) ALPHAR (output) DOUBLE PRECISION array, dimension (N) ALPHAI (output) DOUBLE PRECISION array, dimension (N) BETA (output) DOUBLE PRECISION array, dimension (N) On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i, and BETA(j),j=1,...,N are the diagonals of the complex Schur form (S,T) that would result if the 2-by-2 diagonal blocks of the real Schur form of (A,B) were further reduced to triangular form using 2-by-2 complex unitary transformations. If ALPHAI(j) is zero, then the j-th eigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues are a complex conjugate pair, with ALPHAI(j+1) negative. Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) may easily over- or underflow, and BETA(j) may even be zero. Thus, the user should avoid naively computing the ratio. However, ALPHAR and ALPHAI will be always less than and usually comparable with norm(A) in magnitude, and BETA always less than and usually comparable with norm(B). VSL (output) DOUBLE PRECISION array, dimension (LDVSL,N) If JOBVSL = 'V', VSL will contain the left Schur vectors. Not referenced if JOBVSL = 'N'. LDVSL (input) INTEGER The leading dimension of the matrix VSL. LDVSL >=1, and if JOBVSL = 'V', LDVSL >= N. VSR (output) DOUBLE PRECISION array, dimension (LDVSR,N) If JOBVSR = 'V', VSR will contain the right Schur vectors. Not referenced if JOBVSR = 'N'. LDVSR (input) INTEGER The leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR = 'V', LDVSR >= N. WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) INTEGER The dimension of the array WORK. LWORK >= 8*N+16. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. BWORK (workspace) LOGICAL array, dimension (N) Not referenced if SORT = 'N'. INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. = 1,...,N: The QZ iteration failed. (A,B) are not in Schur form, but ALPHAR(j), ALPHAI(j), and BETA(j) should be correct for j=INFO+1,...,N. > N: =N+1: other than QZ iteration failed in DHGEQZ. =N+2: after reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Generalized Schur form no longer satisfy DELCTG=.TRUE. This could also be caused due to scaling. =N+3: reordering failed in DTGSEN. SEE ALSO INTRO_LAPACK(3S), INTRO_SCSL(3S) This man page is available only online.